Many of the tools used to study classical knots are not available in \(\mathbb {R}P^3\) . For instance, one of the techniques for constructing knots in \(S^3\)  is the notion of connected sum for knots that gives rise to a notion of primeness of a knot. Recall that the connected sum of knots \(K_1\)  and \(K_2\)  in \(S^3\)  is defined using a surgery procedure as : Think of \(K_1\)  and \(K_2\)  to be contained in two distinct copies of \(S^3\) . Now remove an open ball \(B_1\)  from the sphere containing \(K_1\) , which intersects \(K_1\)  at a single arc with its two boundary points on \(\partial \bar{B_1}\) . Similarly remove an open ball \(B_2\)  from the sphere containing \(K_2\) , which intersect \(K_2\)  at a single arc. The complements of \(B_1\)  and \(B_2\)  are closed 3-balls. By gluing their boundaries together, with a diffeomorphism which also sends the boundary points of the remaining arc of \(K_1\)  to the boundary points of the remaining arc of \(K_2\) , we obtain a copy of \(S^3\)  containing a new knot. Note that the ambiguity concerning the choice of points in \(\partial (K_1\setminus B_1)\)  to be identified with specific points in \(\partial ( K_2\setminus B_2)\)  can be taken care if we start with an orientation on both the knots \(K_1\)  and \(K_2\)  [1]}. This gives a well defined notion of \(K_1\#K_2\)  as a new knot in \(S^3\) . This procedure can be performed repeatedly as a sum of an arbitrary number of knots and it is an associative binary operation which defines a monoid structure on the set of knots in \(S^3\) .
